Average Error: 1.6 → 0.1
Time: 41.3s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.429102782305307 \cdot 10^{+43}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 1.4338863214513387 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.429102782305307 \cdot 10^{+43}:\\
\;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 1.4338863214513387 \cdot 10^{-10}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\

\end{array}
double f(double x, double y, double z) {
        double r2404138 = x;
        double r2404139 = 4.0;
        double r2404140 = r2404138 + r2404139;
        double r2404141 = y;
        double r2404142 = r2404140 / r2404141;
        double r2404143 = r2404138 / r2404141;
        double r2404144 = z;
        double r2404145 = r2404143 * r2404144;
        double r2404146 = r2404142 - r2404145;
        double r2404147 = fabs(r2404146);
        return r2404147;
}
↓
double f(double x, double y, double z) {
        double r2404148 = x;
        double r2404149 = -4.429102782305307e+43;
        bool r2404150 = r2404148 <= r2404149;
        double r2404151 = 4.0;
        double r2404152 = r2404151 + r2404148;
        double r2404153 = y;
        double r2404154 = r2404152 / r2404153;
        double r2404155 = z;
        double r2404156 = r2404155 / r2404153;
        double r2404157 = r2404148 * r2404156;
        double r2404158 = r2404154 - r2404157;
        double r2404159 = fabs(r2404158);
        double r2404160 = 1.4338863214513387e-10;
        bool r2404161 = r2404148 <= r2404160;
        double r2404162 = r2404155 * r2404148;
        double r2404163 = r2404152 - r2404162;
        double r2404164 = r2404163 / r2404153;
        double r2404165 = fabs(r2404164);
        double r2404166 = r2404161 ? r2404165 : r2404159;
        double r2404167 = r2404150 ? r2404159 : r2404166;
        return r2404167;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -4.429102782305307e+43 or 1.4338863214513387e-10 < x

    1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied div-inv0.2

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot z\right|\]

    4. Applied associate-*l*0.2

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{x \cdot \left(\frac{1}{y} \cdot z\right)}\right|\]

    5. Simplified0.1

      \[\leadsto \left|\frac{x + 4}{y} - x \cdot \color{blue}{\frac{z}{y}}\right|\]

    if -4.429102782305307e+43 < x < 1.4338863214513387e-10

    1. Initial program 2.4

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied associate-*l/0.1

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]

    4. Applied sub-div0.1

      \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.429102782305307 \cdot 10^{+43}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 1.4338863214513387 \cdot 10^{-10}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))